Properties

 Label 855.1.g.c Level $855$ Weight $1$ Character orbit 855.g Self dual yes Analytic conductor $0.427$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -95 Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,1,Mod(379,855)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("855.379");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$855 = 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 855.g (of order $$2$$, degree $$1$$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$0.426700585801$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.475.1 Artin image: $D_8$ Artin field: Galois closure of 8.2.347236875.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + q^{4} + q^{5} +O(q^{10})$$ q - b * q^2 + q^4 + q^5 $$q - \beta q^{2} + q^{4} + q^{5} - \beta q^{10} + \beta q^{13} - q^{16} - q^{19} + q^{20} + q^{25} - 2 q^{26} + \beta q^{32} - \beta q^{37} + \beta q^{38} + q^{49} - \beta q^{50} + \beta q^{52} + \beta q^{53} - q^{64} + \beta q^{65} + \beta q^{67} + 2 q^{74} - q^{76} - q^{80} - q^{95} - \beta q^{97} - \beta q^{98} +O(q^{100})$$ q - b * q^2 + q^4 + q^5 - b * q^10 + b * q^13 - q^16 - q^19 + q^20 + q^25 - 2 * q^26 + b * q^32 - b * q^37 + b * q^38 + q^49 - b * q^50 + b * q^52 + b * q^53 - q^64 + b * q^65 + b * q^67 + 2 * q^74 - q^76 - q^80 - q^95 - b * q^97 - b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 2 q^{5}+O(q^{10})$$ 2 * q + 2 * q^4 + 2 * q^5 $$2 q + 2 q^{4} + 2 q^{5} - 2 q^{16} - 2 q^{19} + 2 q^{20} + 2 q^{25} - 4 q^{26} + 2 q^{49} - 2 q^{64} + 4 q^{74} - 2 q^{76} - 2 q^{80} - 2 q^{95}+O(q^{100})$$ 2 * q + 2 * q^4 + 2 * q^5 - 2 * q^16 - 2 * q^19 + 2 * q^20 + 2 * q^25 - 4 * q^26 + 2 * q^49 - 2 * q^64 + 4 * q^74 - 2 * q^76 - 2 * q^80 - 2 * q^95

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/855\mathbb{Z}\right)^\times$$.

 $$n$$ $$172$$ $$191$$ $$496$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
379.1
 1.41421 −1.41421
−1.41421 0 1.00000 1.00000 0 0 0 0 −1.41421
379.2 1.41421 0 1.00000 1.00000 0 0 0 0 1.41421
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
95.d odd 2 1 CM by $$\Q(\sqrt{-95})$$
5.b even 2 1 inner
19.b odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.1.g.c 2
3.b odd 2 1 95.1.d.b 2
5.b even 2 1 inner 855.1.g.c 2
12.b even 2 1 1520.1.m.b 2
15.d odd 2 1 95.1.d.b 2
15.e even 4 2 475.1.c.b 2
19.b odd 2 1 inner 855.1.g.c 2
57.d even 2 1 95.1.d.b 2
57.f even 6 2 1805.1.h.b 4
57.h odd 6 2 1805.1.h.b 4
57.j even 18 6 1805.1.o.b 12
57.l odd 18 6 1805.1.o.b 12
60.h even 2 1 1520.1.m.b 2
95.d odd 2 1 CM 855.1.g.c 2
228.b odd 2 1 1520.1.m.b 2
285.b even 2 1 95.1.d.b 2
285.j odd 4 2 475.1.c.b 2
285.n odd 6 2 1805.1.h.b 4
285.q even 6 2 1805.1.h.b 4
285.bd odd 18 6 1805.1.o.b 12
285.bf even 18 6 1805.1.o.b 12
1140.p odd 2 1 1520.1.m.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.1.d.b 2 3.b odd 2 1
95.1.d.b 2 15.d odd 2 1
95.1.d.b 2 57.d even 2 1
95.1.d.b 2 285.b even 2 1
475.1.c.b 2 15.e even 4 2
475.1.c.b 2 285.j odd 4 2
855.1.g.c 2 1.a even 1 1 trivial
855.1.g.c 2 5.b even 2 1 inner
855.1.g.c 2 19.b odd 2 1 inner
855.1.g.c 2 95.d odd 2 1 CM
1520.1.m.b 2 12.b even 2 1
1520.1.m.b 2 60.h even 2 1
1520.1.m.b 2 228.b odd 2 1
1520.1.m.b 2 1140.p odd 2 1
1805.1.h.b 4 57.f even 6 2
1805.1.h.b 4 57.h odd 6 2
1805.1.h.b 4 285.n odd 6 2
1805.1.h.b 4 285.q even 6 2
1805.1.o.b 12 57.j even 18 6
1805.1.o.b 12 57.l odd 18 6
1805.1.o.b 12 285.bd odd 18 6
1805.1.o.b 12 285.bf even 18 6

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(855, [\chi])$$:

 $$T_{2}^{2} - 2$$ T2^2 - 2 $$T_{11}$$ T11

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 2$$
$17$ $$T^{2}$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} - 2$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 2$$
$59$ $$T^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2} - 2$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} - 2$$